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Introduction Codes from graphs Flag-transitive symmetric designs Examples PD-sets for codes related to flag-transitive symmetric designs Nina Mostarac (nmavrovic@math.uniri.hr) Dean Crnkovi c (deanc@math.uniri.hr) Department of

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

PD-sets for codes related to flag-transitive symmetric designs

Nina Mostarac (nmavrovic@math.uniri.hr) Dean Crnkovi´ c (deanc@math.uniri.hr) Department of Mathematics, University of Rijeka, Croatia Supported by CSF (Croatian Science Foundation), Grant 6732 Finite Geometry & Friends A Brussels summer school on finite geometry June 18, 2019

Nina Mostarac (nmavrovic@math.uniri.hr) 1/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Introduction

permutation decoding was introduced in 1964 by

MacWilliams

it uses sets of code automorphisms called PD-sets
the problem of existence of PD-sets and finding them
we will prove the existence of PD-sets for all codes

generated by the incidence matrix of an incidence graph

f a flag-transitive symmetric design and construct some

examples

Nina Mostarac (nmavrovic@math.uniri.hr) 2/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Refrences [1]

D. Crnkovi´

c, N. Mostarac, PD-sets for codes related to flag-transitive symmetric designs, Trans. Comb., 7 (2018) 37–50. [2] P . Dankelmann, J.D. Key and B.G. Rodrigues, Codes from incidence matrices of graphs, Des. Codes Cryptogr., 68 (2013) 373–393.

for prime p let Cp(G) be the p-ary code spanned by the rows of

the incidence matrix G of a graph Γ

we will show that if Γ is the incidence graph of a flag-transitive

symmetric design D, then any flag-transitive automorphism group of D can be used as a PD-set for full error correction for the linear code Cp(G) (with any information set)

Nina Mostarac (nmavrovic@math.uniri.hr) 3/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Codes Definition 1 Let p be a prime. A p-ary linear code C of length n and dimension k is a k-dimensional subspace of the vector space (Fp)n. Definition 2

Let x = (x1, ..., xn) and y = (y1, ..., yn) ∈ Fn
p. The Hamming

distance between words x and y is the number d(x, y) = |{i : xi = yi}|.

The minimum distance of the code C is defined by

d = min{d(x, y) : x, y ∈ C, x = y}.

Notation: [n, k, d]p code
it can detect at most d − 1 errors in one codeword and correct at

most t = d−1

2

errors

Nina Mostarac (nmavrovic@math.uniri.hr) 4/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Graphs

We will discuss undirected graphs, with no loops and multiple edges. Definition 3 Edge connectivity λ(Γ) of a connected graph Γ is the minimum number of edges that need to be removed to disconnect the graph. Remark 1 For every graph Γ: λ(Γ) ≤ δ(Γ).

Nina Mostarac (nmavrovic@math.uniri.hr) 5/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Codes from incidence matrices of graphs

Let G be the incidence matrix of a graph Γ = (V, E) over Fp, p prime and the code Cp(G) the row-span of G over Fp. Theorem 2.1 (Dankelmann, Key, Rodrigues [2](Result 1)) Let Γ = (V, E) be a connected graph and G its incidence

matrix. Then:

1 dim(C2(G)) = |V| − 1; 2 for odd p, dim(Cp(G)) = |V| if Γ is not bipartite, and

dim(Cp(G)) = |V| − 1 if Γ is bipartite.

Nina Mostarac (nmavrovic@math.uniri.hr) 6/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Codes from incidence matrices of graphs

Theorem 2.2 (Dankelmann, Key, Rodrigues [2](Theorem 1)) Let Γ = (V, E) be a connected graph, G a |V| × |E| incidence matrix for G. Then:

1 C2(G) is a [|E|, |V| − 1, λ(Γ)]2 code; 2 if Γ is super-λ, then C2(G) is a [|E|, |V| − 1, δ(Γ)]2 code,

and the minimum words are the rows of G of weight δ(Γ).

Nina Mostarac (nmavrovic@math.uniri.hr) 7/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Codes from incidence matrices of graphs

Theorem 2.3 (Dankelmann, Key, Rodrigues [2](Theorem 2)) Let Γ = (V, E) be a connected bipartite graph, G a |V| × |E| incidence matrix for G, and p an odd prime. Then:

1 Cp(G) is a [|E|, |V| − 1, λ(Γ)]p code; 2 if Γ is super-λ, then Cp(G) is a [|E|, |V| − 1, δ(Γ)]p code,

and the minimum words are the non-zero scalar multiples

f the rows of G of weight δ(Γ).

Nina Mostarac (nmavrovic@math.uniri.hr) 8/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Codes from incidence matrices of graphs Theorem 2.4 (Dankelmann, Key, Rodrigues [2](Result 3)) Let Γ = (V, E) be a connected bipartite graph. Then λ(Γ) = δ(Γ) if

ne of the following conditions holds:

1 V consists of at most two orbits under Aut(Γ), and in particular if

Γ is vertex-transitive;

2 every two vertices in one of the two partite sets of Γ have a

common neighbour;

3 diam(Γ) ≤ 3; 4 Γ is k-regular and k ≥ n + 1

4 ;

5 Γ has girth g and diam(Γ) ≤ g − 1.

Nina Mostarac (nmavrovic@math.uniri.hr) 9/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Information sets

Definition 4 Let C ⊆ Fn

p be a linear [n, k, d] code. For I ⊆ {1, ..., n} let

pI : Fn

p → F|I| p , x → x|I, be an I-projection of Fn

p. Then I is called

an information set for C if |I| = k and pI(C) = F|I|

p .

The set of the first k coordinates for a code with a generating matrix in the standard form is an information set.

Nina Mostarac (nmavrovic@math.uniri.hr) 10/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

PD-sets Definition 5 Let C ⊆ Fn

p be a linear [n, k, d] code that can correct at most t errors,

and let I be an information set for C. A subset S ⊆ AutC is called a PD-set for C if every t-set of coordinate positions can be moved by at least one element of S out of the information set I. A lower bound on the size of a PD-set: Theorem 3.1 (The Gordon bound) If S is a PD-set for an [n, k, d] code C that can correct t errors, r = n − k, then: |S| ≥ n r n − 1 r − 1

· · ·

n − t + 1 r − t + 1

· · ·
.

Nina Mostarac (nmavrovic@math.uniri.hr) 11/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Symmetric designs

Definition 6 A symmetric (v, k, λ)-design is an incidence structure D = (P, B, I) which consists of the set of points P, the set of blocks B and an incidence relation I such that:

|P| = |B| = v,
every block is incident with exactly k points
and every pair of points is incident with exactly λ blocks

(λ > 0). A symmetric (v, k, 1)-design is called a projective plane of order k − 1, and a symmetric (v, k, 2)-design is called a biplane.

Nina Mostarac (nmavrovic@math.uniri.hr) 12/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Incidence graph of a symmetric design

Definition 7 An incidence graph or a Levi graph of a symmetric design is a graph whose vertices are points and blocks of the design, and edges are incident point-block pairs (flags). Remark 2 An incidence graph Γ of a symmetric (v, k, λ)-design:

is bipartite,
is k-regular,
has diameter diam(Γ) = 3.

Nina Mostarac (nmavrovic@math.uniri.hr) 13/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Flag transitive symmetric designs

Definition 8

An automorphism of a symmetric design is a permutation
f points which sends blocks to blocks.
An automorphism group of a symmetric design D is called

flag-transitive if it is transitive on flags of D.

Nina Mostarac (nmavrovic@math.uniri.hr) 14/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Theorem 3.2 (Dankelmann, Key, Rodrigues [2](Result 7)) Let Γ = (V, E) be a k-regular graph with the automorphism group A transitive on edges and let G be an incidence matrix of Γ. If C = Cp(G) is a [|E|, |V| − ε, k]p code, where p is a prime and ε ∈ {0, 1, ...|V| − 1}, then any transitive subgroup of A is a PD-set for full error correction for C. Theorem 3.3 (D.C., N.M.) Let Γ = (V, E) be an incidence graph of a symmetric (v, k, λ)-design D with flag-transitive automorphism group A and let G be an incidence matrix for Γ. Then C = Cp(G) is a [|E|, |V| − 1, k]p code, for any prime p, and any flag transitive subgroup of A can serve as a PD-set (for any information set) for full error correction for the code C.

Nina Mostarac (nmavrovic@math.uniri.hr) 15/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Examples

for the following computational results we use

programming packages GAP and Magma

1 examples of flag-transitive projective planes 2 examples of flag-transitive biplanes

Parameters of the linear [n, k, d]p code obtained from a flag-transitive symmetric (v, k′, λ)-design in the described way are:

n = v · k′
k = 2v − 1
d = k′

Nina Mostarac (nmavrovic@math.uniri.hr) 16/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Flag-transitive projective planes i Flag- Code Gordon Orders of all Smallest transitive Cp(Gi) bound flag-transitive PD-set projective gi subgroups of found plane Di

autom. group Ai

in Ai 1 (7, 3, 1) [21,13,3 ] 3 21,168 4 2 (13, 4, 1) [52,25,4] 2 5616 4 3 (21, 5, 1) [105,41,5] 4 20160, 40320, 64 60480, 120960

Nina Mostarac (nmavrovic@math.uniri.hr) 17/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Flag-transitive biplanes Flag-transitive symmetric Code Gordon Orders of all i design Di, full automorphism Cp(Gi) bound flag-transitive group Ai, point stabilizer gi subgroups of Ai 4 (4, 3, 2), S4, S3 [12, 7, 3] 3 12, 24 5 (7, 4, 2), PSL2(7), S4 [28, 13, 4] 2 168 6 (11, 5, 2), PSL2(11), A5 [55, 21, 5] 4 55, 660 96, 192, 288, 7 (16, 6, 2), 24S6, S6 [96, 31, 6] 3 384, 576, 768, 960, 1152, 1920, 5760, 11520 8 (16, 6, 2), (Z2 × Z8)(S2.4), [96, 31, 6] 3 384, 768 (S2.4)

Nina Mostarac (nmavrovic@math.uniri.hr) 18/20 PD-sets for codes related to flag-transitive symmetric designs

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Introduction Codes from graphs Flag-transitive symmetric designs Examples

Flag-transitive biplanes

i Flag-transitive Code Gordon Smallest PD-set design Di C2(Gi) bound gi found in Ai 4 (4, 3, 2) [12,7,3 ] 3 3 5 (7, 4, 2) [28,13,4] 2 3 6 (11, 5, 2) [55,21,5] 4 10 7 (16, 6, 2) [96,31,6] 3 12 8 (16, 6, 2) [96,31,6] 3 9

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